Question

the bearing of a ship from a lighthouse was found to be n 17° e. after the ship sailed 9.1 miles due south, the new bearing was n 30° e. find the distance between the ship and the lighthouse at each location. the ship began 20.2 miles from the lighthouse. (simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.) the ship finished at □ miles from the lighthouse. (simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)

Explanation:

Step1: Find the angle between the two paths

The initial bearing is \(N30^{\circ}E\) and after sailing south (bearing \(180^{\circ}\) from north), the new - bearing situation. The angle between the initial position of the ship (from the lighthouse) and the final position of the ship (after sailing south) can be found using angle - relation.
The angle at the lighthouse, \(A=30^{\circ}-17^{\circ}=13^{\circ}\). Let the initial distance of the ship from the lighthouse be \(c = 20.2\) miles and the distance the ship sails south be \(b = 9.1\) miles. We want to find the new distance \(a\) from the lighthouse to the ship.
We use the Law of Cosines: \(a^{2}=b^{2}+c^{2}-2bc\cos A\)

Step2: Substitute the values into the Law of Cosines formula

Substitute \(b = 9.1\), \(c = 20.2\), and \(A = 13^{\circ}\) (\(\cos(13^{\circ})\approx0.9744\)) into the formula:

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Answer:

\(11.8\)